Substituting r/t for v or dr/dt?

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1. Feb 11, 2016

Mr Davis 97

I am a little confused about how we define the kinematic quantities that are rates. What is velocity defined as? Is it the instantaneous time rate of change of displacement, or is it simply displacement divided by time? Here is an example of where this problem comes up:

I need to solve the equation $\displaystyle v = \omega r \sqrt{\frac{m_b}{m_s}}$ for time. The only way I see of doing this is substituting $\displaystyle \frac{r}{t}$ for $\displaystyle v$ in order to cancel the two r's and solve for time. However, what gives me the right to make this substitution? Isn't $\displaystyle \frac{r}{t}$ just the average velocity, while $\displaystyle \frac{dr}{dt}$ is the instantaneous velocity? Why am I able to choose either or solely for my purposes at hand?

2. Feb 11, 2016

nasu

What is the actual problem and how these quantities depend on time? Is r a function of time? is omega a function of time?

3. Feb 11, 2016

Isaac0427

Is there acceleration involved?